STRICT LYAPUNOV FUNCTION AND CHETAEV FUNCTION FOR STABILITY/INSTABILITY ANALYSIS OF THE PENDULUM 1

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1 STRICT LYAPUNOV FUNCTION AND CHETAEV FUNCTION FOR STABILITY/INSTABILITY ANALYSIS OF THE PENDULUM Rafael Kelly and Victor Santibañez Dirección de Telemática, CICESE Apdo Postal 65, Adm, Carretera Tijuana Ensenada Km 7 Ensenada, B C, 8, MEXICO e mail: rkelly@cicesemx, Fax: + 5 (646) Instituto TecnológicodelaLaguna Apdo Postal 49, Adm Torreón, Coahuila, 7, MEXICO e mail: vsantiba@itlalagunaedumx, Fax: + 5 (87) Abstract: Stability analysis of nonlinear systems can be carried out by the Lyapunov s first method (linearization techniue) However, by utilizing this approach most insight about the physical system may be lost Alternative approaches are available to exploit system characteristics such as those associated to energy concepts for stability and instability analysis In this paper, with reference to a simple nonlinear system the pendulum, we proposed first, a simple strict Lyapunov function motivated by energy consideration to study asymptotic stability of some euilibria, and second a Chetaev function to analyze instability of the remaining ones A remarkable point is that both functions have a common structure based on the system s energy, and differs only in a single scalar parameter Copyright c 5 IFAC Keywords: Stability, instability, Lyapunov theory, Chetaev function, Nonlinear systems INTRODUCTION This paper considers autonomous systems described by ẋ () where x IR n is the state vector, and the vector field is assumed sufficiently smooth The simplest and basic structure in dynamic systems is the euilibrium Euilibria are related to important meaning in many physical processes Work partially supported by CONACyT and COSNET They are those elements of the state space, say x IR n, where the vector field vanishes, ie f(x ) The Lyapunov stability theory mainly focuses at least as presented in important stability analysis textbooks to study a number of attributes of euilibria such as: stability, asymptotic stability, instability, etc The Lyapunov s theory offers a set of theorems to study these attributes of the euilibria (see eg Vidyasagar (993), Khalil (996)) Most of them resort to the concept of Lyapunov function below (Vidyasagar, 993)

2 Definition Assume that the origin of the state space is an euilibrium of the system (), ie f() A continuous and differentiable function V (x) is a Lyapunov function for the euilibrium x if it is a positive definite function (at least locally) whose time derivative along the system (): V (x) T is nonpositive (at least locally) The basic stability theorem of the Lyapunov s direct method states that if a Lyapunov function exists then the euilibrium is stable (Khalil, 996) A powerful class of Lyapunov function is defined in the following (Santibañez and Kelly, 997) Definition A Lyapunov function for the euilibrium x of the system () is a strict Lyapunov function if the time derivative along the system (): V (x) T is negative definite (at least locally) In other words, strict Lyapunov functions are continuously differentiable positive definite functions (at least locally) whose time derivative along the system trajectories are negative definite (at least locally) For instability analysis, it is convenient to introduce the Chetaev functions as follows Definition 3 A Chetaev function for the euilibrium x of the system () is a continuously differentiable function V (x) such that V () V (x) > at least for some arbitrarily small x 6 such that the time derivative along the system () V (x) T is positive definite (at least locally) This definition obeys to conditions evoked by the Chetaev s instability theorem (Khalil, 996) Indeed, Definition 3 imposes stronger reuirements than those needed in the Chetaev s theorem Based in above definitions, we can summarize the following sufficient conditions for stability and instability extracted from well known theorems of the Lyapunov s stability theory (see eg Vidyasagar (993), Khalil (996)) Theorem Consider system () wheretheorigin x is an euilibrium If there exists a continuously differentiable function V (x) such that V (x) is a Lyapunov function, then the euilibrium is stable V (x) is a strict Lyapunov function, then the euilibrium is asymptotically stable V (x) is a Chetaev function, then the euilibrium is unstable The remaining of the paper is devoted to show the application of this theorem for the stability/instability analysis of a simple nonlinear system where its energy function is exploited to introduce a strict Lyapunov function and a Chetaev function with the novelty of a common structure PENDULUM WITH DAMPING A simple and intuitive nonlinear physical system is the pendulum with damping In normalized form, the unforced process dynamics is given by + +sin() where stands for the pendulum angular position This system can be rewritten in state form as d dt sin() This an autonomous system whose set of euilibria E is given by ½ ¾ nπ E where n is an integer number The stability/instability attributes of these euilibria can be studied straightforward by the Lyapunov s first method, that is, by the linearization approach Notwithstanding, we pay attention to exploit the physical structure of the system to study its stability/instability without linearization We believe that doing this, a systematic procedure may be derived for analysis of more complex physical systems The energy function of the system is composed by the sum of the kinetic energy plus the potential energy cos() The euilibria corresponding to n even (including n ) have common topological stability/instability attributes This fact also applies to

3 those euilibria associated to n odd In virtue of these reasons, it is enough to analyze the euilibria resulting for n andn 3 STRICT LYAPUNOV FUNCTION The euilibrium associated to n isknownto be asymptotically stable One way to prove this fact is by finding a strict Lyapunov function and invoking the Theorem However, first we prove that the euilibrium is stable The natural and classical way is by utilizing the pendulum total energy function V L (, ) +[ cos()] which is locally positive definite Its time derivative along the systems trajectories () yields V L (, ) In virtue of Definition, V L (, ) ualifies as a Lyapunov function, but it is not a strict Lyapunov function because it fails to satisfy that V L (, ) is locally negative definite Therefore, in view of Theorem, wehavetheconclusionthattheeuilibrium associated to n is stable Typically, the asymptotic stability attribute is shown by application of the Krasovskii LaSalle s theorem (Vidyasagar, 993) Instead, in the remaining of this Section we will utilize a strict Lyapunov function Now, in this paper we propose the following function inspired from the total energy function V SL (, )V L (, )+ sin(), () +[ cos()] + sin() where the term sin() has been added We have to check that it is a locally positive definite function To this end, it is sufficient to ensure: first, the function vanishes at the euilibrium Second, its gradient is null at the euilibrium Third, its Hessian evaluated at the euilibrium is positive definite Obviously V SL (, ) by direct substitution The gradient is given by sin()+ cos() + sin() which vanishes at the euilibrium Finally the Hessian reads as cos() sin() cos() cos() Straightforward substitution indicates that the Hessian is positive definite at the euilibrium Fig Function V SL (, ) These arguments imply that function V SL (, ) is locally positive definite Figure depicts function V SL (, ) which vanishes at nπ After easy simplifications, the time derivative of V SL (, ) along the system trajectories () leads to V SL (, ) W (, ) where W (, ) cos() + sin() + sin() It will be proven that W (, ) is a locally positive definite function We proceed by the same stages that those utilized to shown that V SL (, ) isalso locally positive definite First observe that W (, ) Second, the gradient can be written by sin() +sin()cos()+ cos() cos + sin() which vanishes at [ ] T [ ] T On the other hand, the Hessian matrix is H (, ) H (, ) H (, ) H (, ) where H (, ) cos() +cos() sin() sin(), H (, )sin() + cos(), H (, )sin() + cos(), H (, ) cos()

4 V C (z, )V L (z, )+sin(z), (4) Fig Function V SL (, ) The Hessian evaluated at the euilibrium produces which corresponds to a positive definite matrix Therefore, these arguments ensure that W (, ) is a locally positive definite function, thus V SL (, ) is a locally negative definite function The plot of V SL (, ) is shown in Figure ; notice that it is negative and it vanishes only at nπ This means that V SL (, ) ualifies as a strict Lyapunov function, and therefore in agreement with Theorem, the euilibrium associated to n is asymptotically stable As a matter of fact, this strict Lyapunov function applies naturally to prove asymptotic stability of all euilibria corresponding to n even +[ cos(z)] + sin(z) It is worth noticing that above function and the strict Lyapunov function () have the same structure and differs only in a scalar of the third right hand term sin(z) Certainly V C (z, ) is not anymore a locally positive definite function, but it satisfies V C (, ) V C (z, ) > for[z ] T [ ] T with as small as desired, and for [z ] T [z ] T with z arbitrarily small AccordingtoDefinition 3, this means that V C (z, ) is a good candidate to be a Chetaev function It remains to prove that its time derivative is locally positive definite After some tedious by straightforward manipulations, the time derivative of V C (z, ) along the trajectories of system (3) leads to V C (z, ) [cos(z) ] +sin(z) (5) Obviously V C (, ) and V C (z, ) > forz and small enough This implies that V C (z, ) in (4) is a locally positive definite function, hence in virtue of Definition 3 we have that V C (z, ) ualifies as a Chetaev function This can be seen in Figure 3 where the domain in the state space where both V C (z, ) and V C (z, ) arepositiveis shown Observe that inside the circle the time derivative V C (z, ) is positive Finally, the instability attribute of the euilibrium follows from Theorem 4 CHETAEV FUNCTION The euilibrium corresponding to n (and in general n odd) is known to be unstable We shift the euilibrium to the origin by the change of variable z π The system ()canberewritten as d dt z sin(z + π) (3) Thus, the origin [z ] T [ ] T corresponds to euilibrium [ ] T [π ] T According to Theorem, itissufficient to find a Chetaev function to prove that the euilibrium is unstable Motivated by the strict Lyapunov function (), in this paper we propose Fig 3 Chetaev function

5 5 CONCLUSIONS Although stability of physical systems can be studied by utilizing the Lyapunov s stability theory resorting to energy like functions, some times it is not sufficient to establish asymptotic stability Then, typically the Krasovskii LaSalle s theorem is invoked to support the Lyapunov s stability theorem On the other hand, few studies concerning energy like function for study instability are available By means of a simple mechanical system, this paperhaveshowntheanalysisofasymptoticstability by proposing a strict Lyapunov function, thus obviating the Krasovskii LaSalle s theorem On the other hand, motivated by this strict Lyapunov function,wehaveproposedachetaevfunction having the same structure to show instability Indeed, this has been shown on a simple process, notwithstanding, we believe that the ideas can be extended to more general systems, in particular for electromechanical systems 6 REFERENCES Khalil, H K, (996) Nonlinear systems analysis, Prentice Hall, Second Edition Santibañez, V and R Kelly, (997) Strict Lyapunov functions for control of robot manipulators, Automatica, Vol 33, No 4, pp Vidyasagar, M, (993) Nonlinear Systems Analysis, Second Edition, Prentice Hall